Subgroups — C₁₀₁×#14454 #38796605

Order 25856.

1190 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1441	C₂	1	4	yes	maximal
3	C₁₀₁×#1441	C₂	1	2	yes	maximal
4	C₁₀₁×#1436	C₂	1	2	yes	maximal
5	C₁₀₁×#1436	C₂	1	4	yes	maximal
6	C₁₀₁×#1434	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₀₁	C₂²	1	2	yes	—
13	#384×C₁₀₁	C₄	1	8	yes	complemented
14	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
15	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
16	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
17	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
18	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
19	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
20	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
21	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
22	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
24	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
29	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
30	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
31	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
33	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
36	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	16	yes	—
39	C₂×C₄²×C₁₀₁	—	2	4	no	—
40	C₂×C₄²×C₁₀₁	—	4	4	no	—
41	C₂×C₄²×C₁₀₁	—	4	2	no	—
42	C₂×C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
43	#95×C₁₀₁	—	2	8	no	—
44	#95×C₁₀₁	—	2	8	no	—
45	#95×C₁₀₁	—	2	4	no	—
46	#95×C₁₀₁	—	4	2	no	—
47	#95×C₁₀₁	—	4	4	no	—
48	#95×C₁₀₁	—	4	4	no	—
49	#95×C₁₀₁	—	4	2	no	—
50	#95×C₁₀₁	Q₈	1	8	yes	—
51	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₁	—	2	4	no	—
53	C₂²×C₄×C₁₀₁	—	2	1	no	—
54	C₂²×C₄×C₁₀₁	—	2	1	no	—
55	C₂²×C₄×C₁₀₁	—	2	4	no	—
56	C₂²×C₄×C₁₀₁	—	2	2	no	—
57	C₂²×C₄×C₁₀₁	—	2	4	no	—
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	—	2	4	no	—
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	2	no	—
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	—	2	4	no	—
65	C₂²×C₄×C₁₀₁	—	2	4	no	—
66	C₂²×C₄×C₁₀₁	—	2	2	no	—
67	C₂²×C₄×C₁₀₁	—	2	4	no	—
68	C₂²×C₄×C₁₀₁	—	2	2	no	—
69	C₂²×C₄×C₁₀₁	—	2	4	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	—	2	4	no	—
73	C₂²×C₄×C₁₀₁	—	2	4	no	—
74	C₂²×C₄×C₁₀₁	—	2	4	no	—
75	C₂²×C₄×C₁₀₁	—	2	4	no	—
76	C₂²×C₄×C₁₀₁	—	2	4	no	—
77	C₂²×C₄×C₁₀₁	—	2	4	no	—
78	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₀₁	C₂×Q₈	1	8	yes	—
83	C₂²×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
85	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
86	C₄⋊C₄×C₁₀₁	—	4	4	no	—
87	C₄⋊C₄×C₁₀₁	—	4	4	no	—
88	C₄⋊C₄×C₁₀₁	—	4	16	no	—
89	C₄⋊C₄×C₁₀₁	—	4	8	no	—
90	C₄²×C₁₀₁	—	2	8	no	—
91	C₄²×C₁₀₁	—	4	8	no	—
92	C₂³×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
93	C₂³×C₁₀₁	C₂²×Q₈	1	2	yes	—
94	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
95	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
96	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂³×C₁₀₁	H(5,𝔽₂)	1	4	yes	—
99	C₂³×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	4	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	2	4	no	—
106	C₂×C₄×C₁₀₁	—	2	8	no	—
107	C₂×C₄×C₁₀₁	—	2	8	no	—
108	C₂×C₄×C₁₀₁	—	2	4	no	—
109	C₂×C₄×C₁₀₁	—	2	4	no	—
110	C₂×C₄×C₁₀₁	—	2	8	no	—
111	C₂×C₄×C₁₀₁	—	2	8	no	—
112	C₂×C₄×C₁₀₁	—	2	8	no	—
113	C₂×C₄×C₁₀₁	—	2	8	no	—
114	C₂×C₄×C₁₀₁	—	2	8	no	—
115	C₂×C₄×C₁₀₁	—	4	4	no	—
116	C₂×C₄×C₁₀₁	—	4	4	no	—
117	C₂×C₄×C₁₀₁	—	4	4	no	—
118	C₂×C₄×C₁₀₁	—	4	2	no	—
119	C₂×C₄×C₁₀₁	—	4	4	no	—
120	C₂×C₄×C₁₀₁	—	4	2	no	—
121	C₂×C₄×C₁₀₁	—	4	2	no	—
122	C₂×C₄×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	4	4	no	—
124	C₂×C₄×C₁₀₁	—	4	2	no	—
125	C₂×C₄×C₁₀₁	—	4	4	no	—
126	C₂×C₄×C₁₀₁	—	4	4	no	—
127	C₂×C₄×C₁₀₁	—	4	4	no	—
128	C₂×C₄×C₁₀₁	—	4	4	no	—
129	C₂×C₄×C₁₀₁	—	4	4	no	—
130	C₂×C₄×C₁₀₁	—	4	4	no	—
131	C₂×C₄×C₁₀₁	—	4	4	no	—
132	C₂×C₄×C₁₀₁	#117	1	4	yes	—
133	C₂×C₄×C₁₀₁	C₄×Q₈	1	8	yes	—
134	C₂²×C₁₀₁	C₂×#117	1	1	yes	characteristic
135	C₂²×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
136	C₂²×C₁₀₁	#518	1	1	yes	characteristic
137	C₂²×C₁₀₁	#518	1	2	yes	—
138	C₂²×C₁₀₁	#519	1	1	yes	characteristic
139	C₂²×C₁₀₁	#539	1	2	yes	—
140	C₂²×C₁₀₁	#539	1	2	yes	—
141	C₂²×C₁₀₁	#542	1	4	yes	—
142	C₂²×C₁₀₁	#544	1	4	yes	—
143	C₂²×C₁₀₁	#553	1	4	yes	—
144	C₂²×C₁₀₁	#554	1	4	yes	—
145	C₂²×C₁₀₁	#554	1	2	yes	—
146	C₂²×C₁₀₁	#555	1	1	yes	characteristic
147	C₂²×C₁₀₁	Q₈²	1	1	yes	characteristic
148	C₂²×C₁₀₁	#560	1	2	yes	—
149	C₂²×C₁₀₁	#560	1	2	yes	—
150	C₄×C₁₀₁	—	2	4	no	—
151	C₄×C₁₀₁	—	2	8	no	—
152	C₄×C₁₀₁	—	4	4	no	—
153	C₄×C₁₀₁	—	4	4	no	—
154	C₄×C₁₀₁	—	4	8	no	—
155	C₄×C₁₀₁	—	8	4	no	—
156	#14454	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₁₀₁	#2313	1	1	yes	characteristic
158	C₂×C₁₀₁	#2331	1	2	yes	—
159	C₂×C₁₀₁	#2345	1	2	yes	—
160	C₂×C₁₀₁	#2369	1	1	yes	characteristic
161	C₂×C₁₀₁	#2380	1	1	yes	characteristic
162	C₂×C₁₀₁	#2790	1	4	yes	—
163	C₂×C₁₀₁	#2792	1	2	yes	—
164	C₂×C₁₀₁	#2795	1	2	yes	—
165	C₂×#384	C₂×C₁₀₁	1	1	yes	characteristic
166	#1441	C₂×C₁₀₁	1	4	yes	—
167	#1441	C₂×C₁₀₁	1	2	yes	—
168	#1436	C₂×C₁₀₁	1	2	yes	—
169	#1436	C₂×C₁₀₁	1	4	yes	—
170	#1434	C₂×C₁₀₁	1	2	yes	—
171	C₁₀₁	#14454	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
173	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	4	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
176	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
177	C₂²×C₄²	C₂²×C₁₀₁	1	2	yes	—
178	#384	C₄×C₁₀₁	1	8	yes	complemented
179	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
180	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
181	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
182	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
183	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
184	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
185	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
186	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
187	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
188	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
189	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
190	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
191	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
192	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	4	1	no	—
194	C₂×C₄⋊C₄	—	4	4	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₄⋊C₄	—	4	4	no	—
198	C₂×C₄⋊C₄	—	4	1	no	—
199	C₂×C₄⋊C₄	—	4	2	no	—
200	C₂×C₄⋊C₄	—	4	4	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	16	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	4	4	no	—
206	C₂×C₄²	—	4	2	no	—
207	C₂×C₄²	C₂×C₄×C₁₀₁	1	4	yes	—
208	#95	—	2	8	no	—
209	#95	—	2	8	no	—
210	#95	—	2	4	no	—
211	#95	—	4	2	no	—
212	#95	—	4	4	no	—
213	#95	—	4	4	no	—
214	#95	—	4	2	no	—
215	#95	Q₈×C₁₀₁	1	8	yes	—
216	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
244	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
245	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
247	C₂²×C₄	C₂×Q₈×C₁₀₁	1	8	yes	—
248	C₂²×C₄	C₂×Q₈×C₁₀₁	1	4	yes	—
249	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
250	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	16	no	—
254	C₄⋊C₄	—	4	8	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	4	8	no	—
257	C₂³	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
258	C₂³	C₂²×Q₈×C₁₀₁	1	2	yes	—
259	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
260	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
261	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
262	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
263	C₂³	H(5,𝔽₂)×C₁₀₁	1	4	yes	—
264	C₂³	2₋¹⁺⁴×C₁₀₁	1	2	yes	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	#117×C₁₀₁	1	4	yes	—
298	C₂×C₄	C₄×Q₈×C₁₀₁	1	8	yes	—
299	C₂²	C₂×#117×C₁₀₁	1	1	yes	characteristic
300	C₂²	C₂×C₄×Q₈×C₁₀₁	1	2	yes	—
301	C₂²	#518×C₁₀₁	1	1	yes	characteristic
302	C₂²	#518×C₁₀₁	1	2	yes	—
303	C₂²	#519×C₁₀₁	1	1	yes	characteristic
304	C₂²	#539×C₁₀₁	1	2	yes	—
305	C₂²	#539×C₁₀₁	1	2	yes	—
306	C₂²	#542×C₁₀₁	1	4	yes	—
307	C₂²	#544×C₁₀₁	1	4	yes	—
308	C₂²	#553×C₁₀₁	1	4	yes	—
309	C₂²	#554×C₁₀₁	1	2	yes	—
310	C₂²	#554×C₁₀₁	1	4	yes	—
311	C₂²	#555×C₁₀₁	1	1	yes	characteristic
312	C₂²	Q₈²×C₁₀₁	1	1	yes	characteristic
313	C₂²	#560×C₁₀₁	1	2	yes	—
314	C₂²	#560×C₁₀₁	1	2	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	8	no	—
317	C₄	—	4	8	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	4	no	—
321	C₂	C₁₀₁×#2313	1	1	yes	characteristic
322	C₂	C₁₀₁×#2331	1	2	yes	—
323	C₂	C₁₀₁×#2345	1	2	yes	—
324	C₂	C₁₀₁×#2369	1	1	yes	characteristic
325	C₂	C₁₀₁×#2380	1	1	yes	characteristic
326	C₂	C₁₀₁×#2790	1	4	yes	—
327	C₂	C₁₀₁×#2792	1	2	yes	—
328	C₂	C₁₀₁×#2795	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
13. #384×C₁₀₁	C₄	semidirect
156. #14454	C₁₀₁	direct (N × H)
171. C₁₀₁	#14454	direct (N × H)
178. #384	C₄×C₁₀₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#384×C₁₀₁	2
C₁₀₁×#1441	2
C₁₀₁×#1436	2
C₁₀₁×#1434	2
#14454	101

Subgroups — C101×#14454 #38796605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#14454 #38796605